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User blog:B1mb0w/Strong D Function
Strong D Function The strong D function is based on the weaker d function defined in User blog:B1mb0w/Deeply Nested Ackermann. The rules are similar with the significant change being that the D function: \(D(x_1,x_2,x_3,x_4,...,x_n)\) expands to this function: \(D( x_1-1, D(x_1,x_2,x_3,x_4,...,x_n-1), ..., D(x_1,x_2,x_3,x_4,...,x_n-1))\) The same expansion is used to replace each input parameter \(x_2\) to \(x_n\). For 2 parameters, the D function is equivalent to the d function: \(d(a,b)=d(a-1,d(a,b-1))=D(a,b)=D(a-1,D(a,b-1))\) For 3 parameters, the D function quickly dominates the weaker d function: \(d(a,b,c)=d(a-1,d(a,b-1),d(a,b,c-1))\) \(D(a,b,c)=D(a-1,D(a,b,c-1),D(a,b,c-1))\) Calculated Examples \(D() = 0\) This is a null function that always returns zero. \(D(3) = 4\) This is the successor function \(D(1,2) = 5\) This is the same as d(1,2) \(D(1,0,0)\) expands as follows: \(= D(0, D(0,1,1), D(0,1,1)) = D(4,4)\) This is the same as d(4,4) and is comparable to \(f_3(6)\) In fact \(D(4,4) >> f_{\omega}(3)\) \(D(1,0,1)\) expands as follows: \(= D(0, D(1,0,0), D(1,0,0)) = D(D(4,4),D(4,4)) >> D(f_{\omega}(3),f_{\omega}(3))\) and is comparable to \(f_{\omega+1}(3)\) My calculations show that \(D(1,0,n)\) is comparable to \(f_{\omega+n}(3)\) More Examples with 3 parameters \(D(1,1,0) = D(0,D(1,0,1),D(1,0,1)\) which is equal to \(D(1,0,2)\) and comparable to \(f_{\omega+2}(3)\) Similarly \(D(1,1,1) = D(0,D(1,0,2),D(1,0,2)\) which is equal to \(D(1,0,3)\) and comparable to \(f_{\omega+3}(3)\) My calculations show that \(D(1,1,n)\) is comparable to \(f_{\omega+n+2}(3)\) Next \(D(1,2,0) = D(0,D(1,1,2),D(1,1,2)\) which is equal to \(D(1,1,3)\) and comparable to \(f_{\omega+5}(3)\) Similarly \(D(1,2,1) = D(0,D(1,1,3),D(1,1,3)\) which is equal to \(D(1,1,4)\) and comparable to \(f_{\omega+6}(3)\) My calculations show that \(D(1,2,n)\) is comparable to \(f_{\omega+n+5}(3)\) and \(D(1,3,n)\) is comparable to \(f_{\omega+n+8}(3)\) and \(D(1,m,n)\) is comparable to \(f_{\omega+n+(m+1)!^+-1}(3)\) where \(!^+\) is the additive version of factorial. D function examples with 3 parameters - continues \(D(2,0,0)\) will grow significantly faster. \(\) d(1,3,z) >> d(6,z-1) d(1,y,z) >> d(y+3, z-1) d(2,0,z) = d(1, d(1,2), d(2,0,z-1)) >> d( d(1,2)+3 , d(2,0,z-1) -1) >> d(8,z) Graham's Number The d function can reach g1 where g4 is Graham's number relatively quickly. As shown above d(1,3,0) >> g1 . But from my analysis we need the power of the d function with many more input parameters to reach G (Graham's Number). d(6,0) >> g1 d( d(6,0), 0) >> g2 d(6,0,0) = d(5, d(5,6), !!!) where !!! is a big number that we can ignore for now = d(4, d(4, d(5,6)-1), !!!) ... = d(0, d(1, d(2, d(3, d(4, d(5,6)-1)-1)-1)-1)-1), !!!) and the leading parameter = 0 can be dropped. The value of the second parameter is big but does not equal d(6,0) therefore d(6,0,0) << g2 but if we expand d(6,1,0) we will get d(6,0) nested in the long expansion and therefore d(6,1,0) >> g2 by a very great amount. We can reach g3 by moving to 4 input parameters d(6,1,1,0) >> g3 We finally reach G (Graham's Number) with 64 input parameters d(6,1,1,1, ..., 1,0) >> G Growth Rate The growth rate of the d function is much faster than the Fast-growing hierarchy . Here are some comparisons: d(x,y) = d(x-1, d(x,y-1)) = d^(y+2) (x-1,x) fx (y) = f(x-1)^y (y) even if y>>x the d function will exceed f because y+2 > y and the extra function recursion will always generate the largest number. If y >> >> x by a ridiculous amount then this may not be true. and also f6 (6) >> g1 d(6,6) >> f6 (6) >> d(6,0) >> g1 Next My next blog post will introduce a new Alpha function that I have been thinking about. You can find it here: User blog:B1mb0w/Alpha Function Category:Blog posts